I just finished reading Lee Smolin's book The Trouble With Physics: The Rise of String Theory, The Fall of a Science, and What Comes Next. It's really two books in one -- one about the history of modern physics (including, roughly speaking, how the string theorists have taken over) and one about how Smolin believes this is a Bad Idea. Apparently string theory doesn't look so promising these days, and yet the string theorists still dominate physics departments. He's currently at the Perimeter Institute, one of the aims of which is to break down this orthodoxy and give its researchers the freedom to pursue their interests.
Smolin, in examining possible research questions for the future, considers that perhaps it would be a good idea to try to find new interpretations of already surprising facts. For example, consider the length scale implied by the "cosmological constant", which is the length scale over which the universe is curved; this he calls R, which is about 10 billion light-years. What else happens on this scale? The obvious thing is, of course, the universe as a whole. But it turns out that c2/R also seems to be an intersting scale to look at. The constant c2/R (where c is the speed of light) has the units of acceleration, and is about 10-8 centimeters per second. It seems, actually, that Newton's law of gravitation might break down for very small accelerations; this is Mordehai Milgrom's theory of modified Newtonian dynamics (MOND). This can be seen by looking at stars orbiting the center of their galaxies; when the acceleration due to gravity is greater than this critical value, Newtonian gravity seems to work, but when it's less than this critical value it doesn't. When you get below the critical value, it appears that gravitational force falls off only as the inverse distance, not the inverse square distance.
When I read this, though, I was a bit skeptical, because Smolin says that when the acceleration is small, it varies as the square root of mass. This seems to violate everything I know; shouldn't forces be additive? If I'm reading Smolin's version correctly, it's saying that a galaxy four times as heavy should produce an acceleration only twice as large... but only if you view that galactic center as one large object, and not four smaller ones. A bit more ridiculously, consider a galactic center of mass M. It exerts a certain force F on some object far away. But now view that same center as k2 centers of mass M/k2. Then each of those centers exerts a force F/k on the object; the total force on the object is thus k2(F/k) = kF. How does the galactic center "know" whether it's one object or many?
I don't see this criticism anywhere, and it seems so obvious... but I'm not sure whether the issue is with MOND as a theory, with Smolin's summary of it, or with my own understanding of physics.